Set | Course Paper

1

Sets

Set: A set is a collection of well-defined objects. From a set we can say whether a given object belongs to a set

or not.

Those objects are generally called elements of the set. The symbol means ‘is an element of.’

So, it stands to reason that represents ‘is not an element of.’

• Order is immaterial. {x, u, v, t} can be written as {u, t, x, v}

Methods of describing the sets:

There are several ways to notate a set, the two most common ways are:

1. the roster form or Listing method or tabular form and

2. set builder notation, selector method, or rule method

Roster form: Roster form just lists out the elements of a set between two set brackets.

For example, S = {January, June, July}

Set builder notation: Set builder notation describes the members of the set without listing them. This method

does not list the elements but indicate by description of their characteristics. It is also written between two set

brackets.

For example, S = {x | x is a month that begins with J}

Example 1: List the elements of the set, S = {x | x is a natural number between 5 and 11}.

There are two parts to this:

Part 1: the number has to belong to the set of natural numbers {1, 2, 3, …} and

Part2: it has to be between 5 and 11 – which by the way would not include those two numbers.

S= {6,7,8,9,10}

Example 2: List the elements of the set, S = {x | x is a natural number greater than 100 and less than 200}.

There are two parts to this:

1. the number has to belong to the set of natural numbers {1, 2, 3, …} and

2. it has to be greater than 100.

S={101,102,….198,199}

Example 3: Graph the set on a number line. {-2, –.5, 0, 1/5, 3}. Let’s see what we get when we graph all of

these real numbers:

Types of set

Finite set: A set is finite if it contains a finite number of different elements.

Example: S = {1, 2, 3, 4}

Infinite set: Otherwise

Example: S = {1, 2, 3, 4, ………..}

Null set: A set which contains no elements is called a null set or empty set or void set. Empty (or null, or

void) set is a set that contains no elements. It is symbolized by { } OR

• It is real tempting to use them together, but { } IS NOT a way to indicate empty set.

Singleton set: A set which contains a single element is called a singleton set.

Example: S = {4}